# Benjamin Banneker's Trigonometry Puzzle - Explanation of Banneker's Solution

Author(s):
Florence Fasanelli, Graham Jagger, Bea Lumpkin

It is clear that Banneker is using the Law of Sines**: In a triangle, the ratios of the sine of an angle to the length of its opposite side are equal. Where Banneker writes in his proportion, "logarithm base 26" or "logarithm of the hypotenuse," he is anticipating the use of logarithms for the computation. Banneker understood, as his calculations correctly show, that the ratios involved are the sine of an angle to the side opposite, not to the log of the side. In what follows, all angles are expressed in degrees.

I. To find the hypotenuse, Banneker used the Law of Sines: sin C/ c = sin B/ b.

"Sine complement of the angle at A," is sin 60, the sine of the angle complementary to angle A. If x is the length of the hypotenuse, then sin 60/ 26 = sin 90/ x and x = 26 sin 90/ sin 60. Taking logarithms, we have

log x = log 26 + log sin 90 ­­- log sin 60

and, substituting values from a suitable set of tables,

log x = 1.41497 +10 – 9.93753 = 1.47744.

Notice that, for reasons we shall see later, log sin 90 = Log 1010 = 10. We now find x as the antilogarithm of 1.47744, which is very close to 30. It is unlikely that Banneker would have had access to tables of antilogarithms, a late eighteenth century invention, but would simply have used his table of logarithms in reverse.

II. To find the remaining side, Banneker uses the Law of Sines again: sin C/ c = sin A/ a. If x is the length of the side perpendicular to the base then sin 60/ 26 = sin 30/ x, and x = 26 sin 30/sin 60. Taking logarithms we get

log x = log 26 + log sin 30 - log sin 60

and, again substituting values from tables,

log x = 1.41497 + 9.69897 - 9.93753 = 1.7641.

Again, x is the antilogarithm of 1.17643, which is very close to 15.

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**Beatrice Lumpkin’s comments on Banneker’s Trigonometry Puzzle were used for the printed program at a Benjamin Banneker Association session at the NCTM meeting in 1992. See also her article, “From Egypt to Benjamin Banneker: African Origins of False Position Solutions,” in Vita Mathematica: Historical Research and Integration with Teaching, Ronald Calinger, Ed., Washington: MAA, 1996, pp. 279-289. The article is a shortened version of a paper presented at the Quadrennial Meeting of the International Study Group on the Relations Between History and Pedagogy of Mathematics (HPM) at the University of Toronto, August, 1992.

Florence Fasanelli, Graham Jagger, Bea Lumpkin, "Benjamin Banneker's Trigonometry Puzzle - Explanation of Banneker's Solution ," Convergence (June 2010)